Solve \(\sqrt { y+1 } +\sqrt { 2y-5 } \) = 3

Simplyfy
\(\frac { 4{ x }^{ 2 }y }{ 2{ x }^{ 2 } } \times \frac { 6x{ z }^{ 3 } }{ 20{ y }^{ 4 } } \)

Find the values of a and b if the following polynomials are perfect squares
4x⁴ - 12x³ + 37x² + bx + a

If A = \(\left[ \begin{matrix} 1 & 2 & 1 \\ 2 & -1 & 1 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 2 & -1 \\ -1 & 4 \\ 0 & 2 \end{matrix} \right] \) show that (AB)^T = B^TA^T

If α, β are the roots of the equation 2x² - x - 1 = 0, then form the equation whose roots are
2α + β, 2β + α

In \(\triangle\) ABC, if DE||BC, AD = x, DB = x − 2, AE = x +2 and EC = x − 1 then find the lengths of the sides AB and AC.

To get from point A to point B you must avoid walking through a pond. You must walk 34 m south and 41 m east. To the nearest meter, how many meters would be saved if it were possible to make a way through the pond?

PQ is a chord of length 8 cm to a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length of the tangent TP.

In figure, O is the centre of the circle with radius 5 cm. T is a point such that OT = 13 cm and OT intersects the circle E, if AB is the tangent to the circle at E, find the lenght of AB

An artist has created a triangular stained glass window and has one strip of small length left before completing the window. She needs to figure out the length of left out portion based on the lengths of the other sides as shown in the figure.

In the figure, the quadrilateral swimming pool shown is surrounded by concrete patio. Find the area of the patio.

The line joining the points A(0,5) and B(4,1) is a tangent to a circle whose centre C is at the point (4,4) Find the equation of the line AB.

The line joining the points A(0,5) and B(4,1) is a tangent to a circle whose centre C is at the point (4, 4) find The coordinates of the point of contact of tangent line AB with the circle

prove the following identities.
\(\frac { sinA-sinB }{ cosA+cosB } +\frac { cosA-cosB }{ sinA+sinB } =0\)

To a man standing outside his house, the angles of elevation of the top and bottom of a window are 60° and 45° respectively. If the height of the man is 180 cm and if he is 5 m away from the wall, what is the height of the window?(\( \sqrt { 3 } \) = 1.732)

A traveler approaches a mountain on highway. He measures the angle of elevation to the peak at each milestone. At two consecutive milestones the angles measured are 4° and 8°. What is the height of the peak if the distance between consecutive milestones is 1 mile. (tan4° =0.0699, tan8° =0.1405)

Three villagers A, B and C can see each other across a valley. The horizontal distance between A and B is 8 km and the horizontal distance between B and C is 12 km. The angle of depression of B from A is 20° and the angle of elevation of C from B is 30° . Calculate : the vertical height between A and B.(tan20° = 0.3640,(\(\sqrt { 3 } \) = 1.732)

From the top of a tower 50 m high, the angles of depression of the top and bottom of a tree are observed to be 30° and 45° respectively. Find the height of the tree.(\(\sqrt { 3 } \) = 1.732)

Three villagers A, B and C can see each other across a valley. The horizontal distance between A and B is 8 km and the horizontal distance between B and C is 12 km. The angle of depression of B from A is 20° and the angle of elevation of C from B is 30° . Calculate the vertical height between B and C (tan20° = 0.3640,(\(\sqrt { 3 } \)=1.732)

The internal and external radii of a hollow hemispherical shell are 3 m and 5 m respectively. Find the T.S.A. and C.S.A. of the shell.

Calculate the mass of a hollow brass sphere if the inner diameter is 14 cm and thickness is 1mm, and whose density is 17.3 g/ cm³.

A jewel box is in the shape of a cuboid of dimensions 30 cm x 15 cm x 10 cm surmounted by a half part of a cylinder as shown in the figure. Find the volume and T.S.A. of the box.

As shown in figure a cubical block of side 7 cm is surmounted by a hemisphere. Find the surface area of the solid.

An oil funnel of tin sheet consists of a cylindrical portion 10 cm long attached to a frustum of a cone. If the total height is 22 cm, the diameter of the cylindrical portion be 8cm and the diameter of the top of the funnel be 18 cm, then find the area of the tin sheet required to make the funnel.

The measurements of the diameters (in cms) of the plates prepared in a factory are given below. Find its standard deviation.

Diameter(cm)	21-24	25-28	29-32	3-6	37-40	41-44
Number of plates	15	18	20	16	8	7